Abstract Prescribing conformally the scalar curvature of a Riemannian manifold as a given function consists in solving an elliptic PDE involving the critical Sobolev exponent. One way of attacking this problem consist in using subcritical approximations for the equation, gaining compactness properties. Together with the results in [30] , we completely describe the blow-up phenomenon in case of uniformly bounded energy and zero weak limit in positive Yamabe class. In particular, for dimension greater or equal to five, Morse functions and with non-zero Laplacian at each critical point, we show that subsets of critical points with negative Laplacian are in one-to-one correspondence with such subcritical blowing-up solutions.

中文翻译：

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规定莫尔斯标量曲率：亚临界吹胀解决方案

摘要 共形地规定黎曼流形的标量曲率作为给定函数包括求解涉及临界 Sobolev 指数的椭圆偏微分方程。解决这个问题的一种方法是对方程使用亚临界近似，从而获得紧凑性。连同 [30] 中的结果，我们完整地描述了在正 Yamabe 类中均匀有界能量和零弱极限情况下的爆炸现象。特别是，对于大于或等于 5 的维度，莫尔斯函数和在每个临界点具有非零拉普拉斯算子，我们表明具有负拉普拉斯算子的临界点子集与这种亚临界吹胀解一一对应。